Several notions of measurability for multifunctions are introduced and their relations are explained. The main results concern multifunctions \(F\) defined on the product of two separable complete metric spaces, \(T\) and \(X\). The space \(T\) is endowed with the \(\sigma\)-algebra of measurable sets of a Borel measure \(\mu\) and \(X\) with the \(\sigma\)-algebra of its Borel sets. In both main results indicated below, \(F\) is supposed to be Hausdorff measurable in \((t,x)\) and Hausdorff upper semi-continuous in \(x\). One of the main results gives Hukuhara type approximations \(F_n\) of \(F\) that have Carathéodory selections \(f_n:T\times X\to E\), where \(E\) is a separable Banach space and the values of \(F\) are convex closed bounded nonempty subsets of \(E\). Another result is a theorem of Scorza-Dragoni type and it concerns \(F\) with values forming a separable subset of the hyperspace of closed bounded nonempty subsets of a metric space \(Z\).